Analyses of spherical antennas.ppt -...

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Analyses of Spherical antennas

K. W. LeungState Key Laboratory of Millimeter Waves &

Department of Electronic Engineering,City University of Hong Kong

Characteristics of spherical array

Array elements distributed on a sphere surface.

Conformal, low profile, light weight, easy to install on aircraft surfaces

Stable antenna gain and radiation pattern for scanning

Wide angular coverage by activating different arrayelements

Spherical array application

Earth-orbiting scientific satellite system

The user satellite is using the spherical array

[Stockton and Hockensmith 1977]

Stockton & Hockensmith, "Application of spherical arrays -- A simple approach," Antennas and Propagation Society International Symposium, 1977 , vol.15, pp. 202- 205, June 1977. 3

Hemispherical coverage with a moderate gain for steered beams (~13 dBic).

A cluster of 12 patch elements out of a total 120 are activated at a time. Beam steering was accomplished by shifting the active part in small steps.

[Stockton 1978]

Hemispherical patch array for satellite data link:Electronic Switching Spherical Array (ESSA)

2.0-2.3 GHz

551 pre-programedBeams

3-dB Beam width:26°

Max. VSWR: 1.5

R. Stockton, "Electronic switching spherical array antenna", NASA (Unspecified Center), Apr. 1, 1978.4

Elements of the ESSA: RHCP Microstrip Patch Antenna

[Stockton 1978]

Heigth :0.635 cm

Spacing :0.6 - 0.8 λ

Proposed by Mei

Radiate circularly polarizedfields over a wide beamwidth.

Suitable for systems requiringwide-beamwidths (e.g., GPS).

Spherical helical antenna

K. K. Mei and M. Meyer, “Solutions to spherical anisotropic antennas,” IEEE Trans. Antennas Progagat., vol. 12, pp.459-463, 1964.

A. Safaai-Jazi and J. C. Cardoso,"Radiation characteristics of a spherical helical antenna," IEE Proc. Microw. Antennas Propag., vol. 143, no. I, Feb. 1996

Radiation pattern of a 7-turn spherical helical antenna

3dB beamwidths: 60o

Radiation patterns of spherical helical antenna

Measured electric field patterns of the 6-turn cylindrical helical antenna

3dB Beamwidth: ~ 40o

Comparison:radiation pattern of the cylindrical helical antenna

J. D. Kraus,"Helical Beam Antennas for Wide-Band Applications," Proceedings of the IRE,vol. 36, no.10, pp1236-1242,1948. 8

Spherical Solutions

No edge-shaped boundaries as found in cylindrical and rectangular structures

Closed-form Green’s functions obtainable

Exact solutions used as references for checking the accuracy of numerical or approximation techniques

Helmholtz Equation in Spherical Cooridnates

Solution

where bn(kr) is the spherical Bessel function

is the Associated Legendre function of the first kind

is sinusoidal function.

)(cosmnP

jmnmnnm ePkrb )(cos)(,

Remark: Since the associated Legendre function of the second kind, is singular at = 0 or , it is generally not used for engineering EM problems.

)(cosmnQ

0sin1sin

sin11 2

Spherical Bessel Functions

)( 2/1 krBkr

krb nn

where is the ordinary (cylindrical) Bessel function

)(2/1 krBn

)(),(~)( krykrjkrb nnn

Riccati-Bessel Functions (Schelkunoff-typeSpherical Bessel Function)

• All EM fields can be found from 2 potential functions

• Define and

• But Ar, Fr, are not solutions of Helmholtz equation

• Instead, Ar/r, Fr/r are solutions of Helmholtz equation

• Define Riccati-Bessel function

rAA r ˆ

rFF r ˆ

)()(ˆ2/1 krBkrkrkrbkrB nnn

General Solutions of Sphericl PotentialFunctions

mjmnnrr ePkrBFA )(cos)(ˆ~,,

Legendre Functions

m = 0 m = 1

E & H fields in Electromagnetics

Grounded Spherical Hemispherical Dielectric Resonator Antenna: Embedded Magnetic Source

(a) Side View (b) Top View

')(ˆ)'('ˆ')(ˆ)'('ˆ

)(cos)2(

0 rrkrJkrHrrkrHkrJePAG

nnjmmn

arrkHCarkrJBePG

)(ˆ)(ˆ

)(cos0

')(ˆ)'(ˆ')(ˆ)'(ˆ

)(cos)2(

0 rrkrJkrHrrkrHkrJePDG

nnjmmn

arrkHFarkrJEePG

)(ˆ)(ˆ

)(cos0

: Associated Legendre function of the first kind (order m,degree n)

: Spherical Bessel function of the first kind (Schelkunoff type)

: Spherical Hankel function of the second kind (Schelkunoff type)

P xnm( ) ( )J xn

( )( )H xn2

Electric & Magnetic Green’s Functions Due to M

Particular solutions obtained by matching theboundary conditions at the source point (r = r’)

0)(cos

)!()!(

4' ddePMm

nnnrA jmm

ddePddM

krD jmm

)(cossin)!()!(

For a point source

sin')'()'(

')(ˆ)'('ˆ')(ˆ)'('ˆ

)'(sin)(cos)'(cos'sin'

1 1 rrkrJkrHrrkrHkrJmPPa

')(ˆ)'(ˆ')(ˆ)'(ˆ

)'(cos)(cos)'(cos''

1 0 rrkrJkrHrrkrHkrJmPP

nnnjanm

)!()!(

2 mnmn

0 for 20 for 1

Homogenous solutions obtained by matching theboundary conditions at the dielectric surface ( r = a)

,0 EE ,0

HH 0 HH

arrkHcarkrJbkrJmPP

)(ˆ)(ˆ

)'('ˆ)'(sin)(cos)'(cos'sin'

arrkHfarkrJekrJmPP

)(ˆ)(ˆ

)'(ˆ)'(cos)(cos)'(cos''

)(ˆ)('ˆ)('ˆ)(ˆ0

)2()2()2()2( akHkaHkkakHkaHab nno

onnTEn

)('ˆ)(ˆ)(ˆ)('ˆ0

)2()2()2()2( akHkaHkkakHkaHde nno

onnTMn

Magnetic Green’s Function Due to Mr

sin)'()'()'()( 2

')(ˆ)'(ˆ')(ˆ)'(ˆ

)'(cos)(cos)'(cos'1

0 02 rrkrJkrH

rrkrHkrJmPPgr

Particular solution

)12()!()!(

Homogenous solution

arrkHiarkrJhkrJmPP

r )(ˆ)(ˆ

)'(ˆ)'(cos)(cos)'(cos'1

,)(ˆ)('ˆ)('ˆ)(ˆ0

)2()2(

akHkaHkkakHkaHgh nnnnTE

Define 3 Dyadic Green’s Function

rGrGrrGG HM

rr ˆˆˆˆˆˆ

rGrGrrG HM

rˆˆ,ˆˆ,ˆˆ

ˆˆˆˆˆˆ H

HM GGrGG r

')]ˆ'()ˆ'([ dSMGrMGHHM

Total H-field due to Mr & M is given by

'ˆ)''(ˆ)''(ˆ)''(

')ˆ'ˆ'ˆ'()ˆ'ˆ'ˆ'(

dSMGMGMGMGrMGMG

dSMGMGrMGMGMGrMGH

cossin HHH ry Since

'sin'' yr MM

'cos'' yMM

dSMGGGGH

'']cos)'cos'sin(sin)'cos'sin[(

cos)'cos'sin(sin)'cos'sin( H

HM GGGGG

we have

Therefore, the required Green’s function is given by

HPHM GGG y

Expressing the Green’s function in GP & GH:

)()'()'(cos)(cos)'(cos)1('sin'sin1

mnmP krkrmPPgnn

)()'(')'(sin)(cos)'(cos)1('

cos'cos1n

mnm krkrmPPann

)(')'()'(sin)(cos)'(cos'

cos'sin-n

mnm krkrmPPmg

)()'()'(cos)(cos)'(cos''

cos'cos1-n

mnm krkrmP

)(')'(')'(cos)(cos)'(cos'cos'cos

mnm krkrmPPma

'),'(ˆ'),'(ˆ

)'( )2( rrkrHrrkrJkr

nnin which

'),(ˆ'),(ˆ

rrkrJrrkrHkr

)(ˆ)'(ˆ)'(cos)(cos)'(cos)1('sin'sin1

mnmH krJkrJmPPhnn

)(ˆ)'('ˆ)'(sin)(cos)'(cos)1('

cos'cos1n

mnm krJkrJmPPbnn

)('ˆ)'(ˆ)'(sin)(cos)'(cos'cos'sin-

mnm krJkrJmPPmh

)(ˆ)'(ˆ)'(cos)(cos)'(cos''

cos'cos1-n

mnm krJkrJmP

)('ˆ)'('ˆ)'(cos)(cos)'(cos'cos'cos

mnm krJkrJmPPmb

Converting double summations to single summation

By applying the addition theorem for Legendre polynomials

)(cos)(cos)(cos)!()!(2)(cos

mn mPP

)cos(sinsincoscoscos

HPHM GGG y

where we have used the fact that = =/2.

)()'())(cos()12)(1('sin'sin

krkrPnnnrrk

G nnnn

)()'('))(cos('

sin'cos4

krkrPnrr nnn

)(')'())'(cos()12('cos'sin

krkrPnrr nnn

)()'())'(cos(')1(

12'cos'cos

krkrPnnn

rrk nnnn

)(')'('))'(cos('')1(

12'cos'cos

krkrPnnn

)(ˆ)'(ˆ))'(cos()12)(1('sin'sin

krJkrJPnnnbrrk

G nnnn

)(ˆ)'('ˆ))'(cos('

sin'cos4

krJkrJPnbrr nnn

)('ˆ)'(ˆ))'(cos('

)12('cos'sin

krJkrJPnbrr nnn

)(ˆ)'(ˆ))'(cos(')1(

12'cos'cos

krJkrJPnnne

rrk nnnn

)('ˆ)'('ˆ))'(cos('')1(

12'cos'cos

krJkrJPnnnb

)(ˆ)('ˆ)('ˆ)(ˆ1 )2()2()2()2( akHkaHkkakHkaHb onno

onnTEn

)('ˆ)(ˆ)(ˆ)('ˆ1 )2()2()2()2( akHkaHkkakHkaHe onno

onnTMn

Numerical Problem for GP

- GP is an infinite summation over n- Hankel functions of very high orders have very large

ampliudes- Difficult to handle numerically

Recall that physically Gp represents a z-directed electricfield excited by a z-directed point current, therefore

Solution

22 )'sin'()'cos'cos( ryrrR

Mathematical Identity

Based on this fact, a mathematical identity can be established:

yj jkR

)()'())'(cos()12)(1('sin'sin

krkrPnnnrrk nnn

)()'('))(cos('

sin'cos4

krkrPnrr nnn

)(')'())(cos()12('cos'sin

krkrPnrr nnn

)()'())'(cos(')1(

12'cos'cos

krkrPnnn

rrk nnnn

)(')'('))'(cos(')1(

12'cos'cos

krkrPnnn

))'sin'()'cos'cos(( 22 ryrrR

)(ˆ)'(ˆ))'(cos()12)(1('sin'sin

krJkrJPnnnbrrk nnn

)(ˆ)'('ˆ))'(cos('

sin'cos4

krJkrJPnbrr nnn

)('ˆ)'(ˆ))'(cos()12('cos'sin

krJkrJPnbrr nnn

)(ˆ)'(ˆ))'(cos(')1(

12'cos'cos

krJkrJPnnne

rrk nnnn

)('ˆ)'('ˆ))'(cos('')1(

12'cos'cos

krJkrJPnnnb

Finally, the required Green’s function is given as follows:

Method-of-Moments Solution

Expand the equivalent magnetic current of the slot:

)()(),( npun yyfxfyxf

WxWxfu

nnn yfVyM

yhkyf e

)(sin)(

amn YYY

')','(),(2S S

mn dSdSyxfGyxfY

MoM Admittance of the Grounded HemisphericalDielectric Resonator Antenna

')','(),(2S S

mn dSdSyxfGyxfY

However, GP is singular

Difficult to integrate numerically pmnY

Formulation of the Microstrip Feed network- Apply the reciprocity approach as done by David Pozar* - Not repeated here.

D. M. Pozar, “A reciprocity method of analysis for printed slot and slot-coupled microstrip antennas,” IEEE Trans. Antennas Progagat., vol.34, pp. 1439-1446, Dec. 1986.*

Solution: Use the reduced kernel (Richmond form)

2222252 ')'()32)(1()(

L npeeeeee

mn dydyyyfkaajkeyyfk

22)'( ee ayy

with being the equivalent radius of the slot.

Results

At the reference plane (slot position): a = 12.5 mm, ra = 9.5, xd = yd = 0.0, W = 0.9 mm, Wf= 1.45 mm, d = 0.635 mm, rs = 2.96, Ls = 13.6 mm .

At the reference plane (slot position): a = 12.5 mm, ra = 9.5, L = 13.5mm, yd = 0.0, W = 1.3mm, Wf = 1.45 mm, d = 0.635 mm, rs = 2.96, Ls = 13.6 mm .38

At the reference plane (slot position): a = 12.5 mm, ra = 9.5, L = 13.5mm, xd = 0.0, W = 1.3mm, Wf = 1.45 mm, d = 0.635 mm, rs = 2.96, Ls = 13.6 mm .39

Grounded Spherical Hemispherical Dielectric Resonator Antenna: Surface Electric Source

Ground Plane

HemisphericalDRA

Conducting Conformal Strip

r CoaxialAperture

Advantages of Conformal-Strip Method

Compatible with the probe-feed method Drilling hole not required Very convenient post manufacturing trimmings

- Cut shorter without leaving an air hole- Extended longer without the need for

deepening the hole It is conformal

Hemispherical DRA for Demonstration

Analytical closed-form Green function can be obtained

Efficient in numerical implementation Excited in the fundamental broadside TE111

Co-ordinate system in the analysis

Spherical DRA

Conformalr

DRA Green’s FunctionInside the DRA (r < a)

1 )(cos)(ˆn

jmmnnnmr ePkrJAF

1 )(cos)(ˆn

jmmnnnmr ePkrJBA

Outside the DRA (r > a)

2 )(cos)(ˆn

jmmnnnmr ePrkHCF

2 )(cos)(ˆn

jmmnnnmr ePrkHDA

Boundary Conditions

At the DRA-air interface:

0)(ˆ 12 EEr

ˆˆ 12 sJHHr

where Js is the conformal strip current.

On the DRA surface, we have r = r = a and G1 = G2 = G,which is given by:

)'(cos'sin

)'(cossin

)(cos)!()!(2

)('ˆ)('ˆ)1(12

)'(cos'

)'(cos)(cos)!()!(2

)(ˆ)(ˆ)1(12

)2(TM2

)2(TE2

akHkaJnn

dPmnmn

akHkaJnn

)(ˆ)('ˆ)('ˆ)(ˆ0

)2(TE akHkaJkkakHkaJ nnnnn

)('ˆ)(ˆ)(ˆ)('ˆ0

)2(TM akHkaJkkakHkaJ nnnnn

0,20,1

m and 0 is the wave impedance in vacuum.

MoM Solution for the Strip Current

Using the MoM:

qqq fII

)(sin)(

in which

qhLaq 02)1( kk re

(16-18)

MoM Solution for the Strip Current (Cont’)

By Galerkin’s procedure, the following matrix equation is obtained:

)]0([][][ pqpq fIZ where

dSdSfGfW

qppq ')'()',';,()(1

The input impedance is given by

q qq fIZ

Problems in Evaluating Zpq

The Green function G is singular as Excessive no. of modal terms required Higher-order Hankel functions difficult to

handle numerically Considerable computation time required

Solution

),,(),()!()!(2)('ˆ)('ˆ

),,(),()!()!(2)(ˆ)(ˆ

mqpmnmnmnmakHkaJ

mqpmnmnmnakHkaJ

m mTEn

Integral Zpq evaluated using novel recurrence formulas.

First express Zpq in the following form:

,sin)(cos),( 2

dPmn m

)(cos),(2

,')'()'(cos)(),,('1

ddfmfmqp qp

h = h/a.

Recurrence formulas for 2(n,m)(A) Recurrence formula recursive in n

Initial values : 2(0,0) = 2 sin-1x12(1,0) = 0

2(n,0) can be found for all n.

(B) Recurrence formula recursive in m

),())(1()(])1(1[2)2,( 211

2 mnmnmnxPmn mn

),1()()(1)1(1)12()1)(1(

1),1( 21212 mnmnnxPxn

mnnmn m

Initial values : 2(n,0) [from (A) above]2(n, 1) = [(-1)n - 1] Pn(x1)

2(n, m) can be found for all m and n.

Evaluation of 2(n,m) and 1(p,q,m)

),1()1(),1()(12

1)(11)1(),( 221211 mnnmmnmn

nxPxmn m

1(n, m) are found in terms of 2(n, m):

Evaluation of 1(n,m)

Analytical evaluation of 1(p,q,m)

]cos[sin

coscos2

')'()'(cos)(),,(

makmakmhk

ddfmfmqp hp

Radiation Fields

),( qE),( qEwhere and are found in a similar fashion.

Convergence Check

2.8 3 3.2 3.4 3.6 3.8 4 4.2-100

Frequency (GHz)

Input impedance ( )

n=1 to 10

n=1 to 20

n=1 to 40

n=1 to 60

n=1 to 80

a = 12.5 mm, r = 9.5, l = 12.0 mm, and W = 1.2 mm.

Measured and Calculated Input Impedances

2.8 3 3.2 3.4 3.6 3.8 4 4.2

Frequency (GHz)

Input impedance ( )

TheoryExperiment

a = 12.5 mm, r = 9.5, l = 12.0 mm, and W = 1.2 mm

•Measured resonant frequency (zero reactance):3.60 GHz

•Calculated resonant frequency (zero reactance):3.56 GHz

•Error :1.1 %

Results

l = 12 mm

10.9 mm

2.8 3 3.2 3.4 3.6 3.8 4 4.2-50

Frequency (GHz)

Input resistance ( )

2.8 3 3.2 3.4 3.6 3.8 4 4.2-50

Frequency (GHz)

Input reactance ( )

l = 12 mm

10.9 mm

a = 12.5 mm, r = 9.5, and W = 1.2 mm

Modified Configuration with a Parasitic Patch

Grounded parasitic patch

Hemispherical DRA

Conformal excitation strip

Coaxial apertureGround plane

Modified Configuration with a Parasitic PatchSuperscript A : excitation strip Superscript B : parasitic patch

E-field vanishes on the excitation strip

A EEEE

In terms of Green functions

AEJ ESdJGSdJGSdJG

Current expansions of Excitation & Parasitic Patches

1 1)()(),(

B gfII

)()(),(N

B fgII

elsewhere,0

||,sin

|)|sin(~)(),( hagf p

where and are of the following form: )(Blf )(ng

New Integrals of Pnm(cos )

dfPmnp pm

)()(cos),,(1

dPmnp p

)(sin)(cos),,(2

Their recurrence formulas have also been found [A] but are not included here for brevity.

[A] K. W. Leung, and H. K. Ng, “Theory and experiment of circularly polarized dielectricresonator antenna with a parasitic patch,” IEEE Trans. Antennas Propagat., vol. 51, pp.405-412, Mar. 2003.

Measured and Calculated Results

3 3.1 3.2 3.3 3.4 3.5 3.6

Frequency (GHz)

nce Rigorous Theory

Experiment

Resistance

Reactance

Simplified Thoery

a=12.5mm, =9.5, l1=14mm, l2=7.9mm, W1=1.2mm, W2=2.2mm, and o=157.4o.r

Measured and Calculated Results (Cont’d)

a=12.5mm, =9.5, l1=14mm, l2=7.9mm, W1=1.2mm, W2=2.2mm, and o=157.4o.r

Measured and Calculated Results (Cont’d)

f = 3.52 GHzo

(a) x-z plane (b) y-z Plane

90-40 -30 -20 -10 0

-40 -30 -20 -10 0

9090 oo

Right Hand

Left Hand

Right Hand

Left Hand

TheoryExperiment

The method can be used to formulatemicrostrip antenna problems.

Analysis of Spherical Slot Antenna

Perspective View Top View65

ce GGG ,,,

)'(cos)'(cos)(cos)!()!(2)('ˆ)('ˆ

sinsin1

)'(cos'

)'(cos)(cos)!()!(2)(ˆ)(ˆ

mncencen

mcencen

mPPmnmnmakHakJ

dPmnmnakHakJ

)'(cos)'(cos)(cos)!()!(2)('ˆ

sinsin1

)'(cos'

)'(cos)(cos)!()!(2)(ˆ

22)2(2

mPPmnmnmakH

dPmnmnakH

)'(cos)'(cos)(cos)!()!(2

)1(12)('ˆ)(ˆ)('ˆ)(ˆ

sinsin1

)'(cos'

)'(cos)(cos)!()!(2

)1(12)(ˆ)('ˆ)(ˆ)('ˆ

)2()2(2

mPPmnmnm

nnnakHakJcakJakHb

dPmnmn

nnnakHakJfakJakHe

mcncnncncnn

Green’s Function of the Antenna

MoM Admittances

dSdSfGfW

qceHPp

ceHpqP ')'()(1

dSdSfGGfW

ppq ')'()(1

)('sin)(

02)1(' kk re

epqPpq YYYYY

Method A

)]'(sin)33(

)'cos()1([4)(1 2

RjkRka

RjkRkRRk

cecece

),,(),()!()!(2)(

mqpmnmnmnmnB

mqpmnmnmnnA

m mpqH

where2

122 ]2/)'[(sin4 raR

)(ˆ)('ˆ)(ˆ)('ˆ1)(ˆ)( )2()2(2)2( akHbkJfakJakHeakHnA cncnncncnnc

)('ˆ)(ˆ)('ˆ)(ˆ1)('ˆ)( )2()2(2)2( akHbkJcakJakHbakHnB cncnncncnnc

Numerical integration

Recurrence formulas

Method B

• Combine the particular and homogeneous solutions

• Only recurrence formulas are used

• Advantage: Easy to implement and fast (no numerical integration required)

• Disadvantage: ??

Convergence Check for Method A

a = 6.25 cm, b = 3 cm, L = 12.46 cm, W = 2.4 mm, and r = 1.

Convergence Check for Method B

a = 6.25 cm, b = 3 cm, L = 12.46 cm, W = 2.4 mm, and r = 1.

Comparison between Methods A & BAdvantages of Method A

• Method A converges much faster than Method B• More suitable for problems of larger spherical sizesAdvantages of Method B

• Numerical integrations not required• Easier to implement• Much faster than Method A

Method A Method BNumerical

integration for both Gp, GH

32 sec. 112 sec. 77 sec.

Measured and calculated input impedances of the rectangular slot for a = 6.25 cm, b= 4.0 cm, L = 12.46 cm, W = 2.4 mm, and r = 1.

Measured & Calculated (Method B) Results

Recurrence Formulas for Integral of )(ˆ krJ n

Side View Top View

MoM Admittance

, )()()12)(1(2

pqH qApAnnnk

1 sin)(sin)(ˆ

ienn dy

hkyyhk

ykyJiA

)(ˆ)(

)()12)(12(

)2)(1()(])32)(12(

)322([)()32)(12(

)3)(2(

nniAnn

nnkkiA

Recurrence formula for An(i)

Define YD & YC as the dielectric & cavity admittances, respectively.

Backward recurrence for An(i), needless to know A0, A1, A2, & A3.

a = 12.5 mm, r1 = 9.5, L = 20.0 mm, W = 1.0 mm, = 0, and N = 5

Input resistance Input reactance

Solving Planar annular problem usingspherical solution

Top View Side View

MoM Solution

• First obtain the spherical Green’s function of Hdue to a -directed magnetic point current

• Substitute = = /2 (both source & field one the ground plane)

)()()0()1(

)()()0(1)!1()!1(

nnPqnqn

even)(642)1(531)1()0(

MoM Solution (Cont’d)

q qqYZ

in which

otherwise112

a = 19.25 mm, b = 10.75 mm, W = 1.5 mm, r1 = 4, and r2 = 1

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